\[ \boxed { \left ( {x}^{2}-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +a \left ( \left ( y \left ( x \right ) \right ) ^{2}-2\,xy \left ( x \right ) +1 \right ) =0} \]
Mathematica: cpu = 0.086011 (sec), leaf count = 158 \[ \left \{\left \{y(x)\to \frac {\left (x^2-1\right ) \left (c_1 \left (a x \left (x^2-1\right )^{\frac {a}{2}-1} P_{a-1}(x)+\left (x^2-1\right )^{\frac {a}{2}-1} (a P_a(x)-a x P_{a-1}(x))\right )+a x \left (x^2-1\right )^{\frac {a}{2}-1} Q_{a-1}(x)+\left (x^2-1\right )^{\frac {a}{2}-1} (a Q_a(x)-a x Q_{a-1}(x))\right )}{a \left (c_1 \left (x^2-1\right )^{a/2} P_{a-1}(x)+\left (x^2-1\right )^{a/2} Q_{a-1}(x)\right )}\right \}\right \} \]
Maple: cpu = 0.188 (sec), leaf count = 231 \[ \left \{ y \left ( x \right ) ={\frac {1}{ \left ( 4+4\,x \right ) a} \left ( 8\, \left ( \left ( a-1/2 \right ) x-a/2+1/2 \right ) \left ( 1+x \right ) {\it \_C1}\,{\it HeunC} \left ( 0,-2\,a+1,0,0,{a}^{2}-a+1/2,2 \, \left ( 1+x \right ) ^{-1} \right ) -a \left ( -{\frac {x}{2}}-{\frac { 1}{2}} \right ) ^{-2\,a+1} \left ( 1+x \right ) {\it HeunC} \left ( 0,2\,a -1,0,0,{a}^{2}-a+{\frac {1}{2}},2\, \left ( 1+x \right ) ^{-1} \right ) - 8\, \left ( x-1 \right ) \left ( {\it HeunCPrime} \left ( 0,-2\,a+1,0,0,{ a}^{2}-a+1/2,2\, \left ( 1+x \right ) ^{-1} \right ) {\it \_C1}-1/4\, \left ( -x/2-1/2 \right ) ^{-2\,a+1}{\it HeunCPrime} \left ( 0,2\,a-1,0,0 ,{a}^{2}-a+1/2,2\, \left ( 1+x \right ) ^{-1} \right ) \right ) \right ) \left ( {\it HeunC} \left ( 0,-2\,a+1,0,0,{a}^{2}-a+{\frac {1}{2}},2\, \left ( 1+x \right ) ^{-1} \right ) {\it \_C1}-{\frac {1}{4} \left ( -{ \frac {x}{2}}-{\frac {1}{2}} \right ) ^{-2\,a+1}{\it HeunC} \left ( 0,2 \,a-1,0,0,{a}^{2}-a+{\frac {1}{2}},2\, \left ( 1+x \right ) ^{-1} \right ) } \right ) ^{-1}} \right \} \]